Glen A. Hagemann and Joseph R. Cummins
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matrix—A rectangular array of objects. The objects are arranged in rows and columns.
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matrices—The plural of matrix.
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entry—Each number or variable within the matrix.
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dimension or size—The number of rows by the number of columns of the matrix. The number of rows is always given first.
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row matrix—A matrix having one row.
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column matrix—A matrix having one column.
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square matrix—A matrix having the sam e number of rows and columns.
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zero matrix—A matrix where each entry is a zero.
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identity matrix—A matrix where the entries in the main diagonal are ones and all the other entries are zeros.
A magic square is simply a square that possesses equal amounts of boxes in each row an d column. A three by three magic square, for example, has three boxes per row and three boxes per column. There are nine boxes altogether. The objective of a 3 x 3 magic square is to take the natural numbers 1 through 9, and place them into the magic square in such a way that all rows, columns, and corner to corner diagonals have equivalent sums. This sum is called the constant for the square.
Two basic types of magic squares will be discussed. They are odd-order magic squares and even-order magic squares. Odd order magic squares are 3 x 3 matrices, 5 x 5 matrices, 7 x 7 matrices and so on. They are defined as any magic square with an odd number of boxes in each row and in each column. Even order magic squares are those matrices with an even number of boxes in each row and in each column. Examples of even ordered magic squares are 4 x 4 matrices, 6 x 6 matrices, 8 x 8 matrices and so on.
The first step in solving the magic square is to identify the constant. Once the constant is known, it is possible to complete the puzzle by trial and error. Three methods are presented for determining the constant of a 3 x 3 magic square.These methods may be generalized to all odd-order magic squares. (see lesson #1)
Once a certain amount of frustration has been experienced through trial and error, precise and accurate methods may be learned. They are presented as staircase methods A and B. (see lesson #2)
When working with even-order magic squares, one simple method cannot be developed. The patterns and solutions of each individual magic square are dependent upon which natural numbers the magic squares are multiples of. For example, the solutions of a 4 x 4 matrix and a 6 x 6 matrix are not similar because 4 is a multiple of both 2 and 4 while 6 is a multiple of 2,3 and 6. Although 2 is a factor common to both 4 and 6, all other numbers are not. Their differences are enough to change the solutions of the 4 x 4 and 6 x 6 magic squares.
